During the past decades, the exploration of new topological material and the study of their novel physical properties have become a hot topic in condensed matter physics. However, it is hard to realize various topological materials and observe their physical properties that have been predicted theoretically due to the limitation of experimental techniques, such as fabrication, parameter control, and measurement. This situation makes quantum simulation a way alternative to simulating large quantum systems. In general, quantum simulation can be implemented by some controllable quantum systems. As a kind of all-solid state device, superconducting quantum circuit is an artificial quantum system that has great advantage in scalability, integration, and controllability, which provides an important scheme to realize the quantum simulator. In this paper, we review our recent results of quantum simulation in the space-time inversion symmetry protected topological semimetal bands, Hopf-link semimetal bands, and topological Maxwell metal bands with superconducting quantum circuits. These results show that the superconducting circuit is a promising system for simulating the quantum many-body system in condensed matter physics.

Project supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301802) and the National Natural Science Foundation of China (Grant Nos. 11274156, 11504165, 11474152, 61521001).